Chapter 1 Introduction Introduction
4.4 Numerical Simulation and Experimental Validation
4.4.1 Validation of DOB-SMC Using SMP I
The position and orientation tracking control of the SMP are demonstrated from numerical simulations with the comparison of PID, an integral sliding mode control (I-SMC), and proposed DOB- SMC. The results of the estimation of kinematic uncertainties acquired from DOB are analyzed by comparison between simulations and experiments. Moreover, the tracking performance of the DOB- SMC is compared to the PID and I-SMC controller utilizing the prototyped SMP in experiments. For the experiment, the full cockpit sphere is manufactured using glass fiber reinforced plastic compared to the hemisphere in Fig. 2.11(a) to satisfy the full rotational range along all axes, as shown in Fig. 4.4(a).
Due to manufacturing defects, the cockpit sphere has slightly distorted top and bottom surfaces and becomes ellipsoid, as shown in Fig. 4.4(a). In spite of kinematic and dynamic uncertainties caused by a manufacturing error, the DOB-SMC can accomplish the desired trajectory tracking of the cockpit sphere in the experiments. The mechanical parameters for orientation and position control are detailed in Table 4.1.
The SMP requires six actuators to control both translation and rotation. The translational motion can be generated from three rotary actuators and ball screw actuators. The ball screw actuator has a lead γ1, linear displacement according to the rotation of the ball screw. Three rotary actuators are utilized to rotate each small spheres for the rotational motion of the cockpit, and three gearboxes are installed in front of each actuator. The gear ratio of the gearbox is γ2, shown in Table 4.1. According to the
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transmission of force and torque using the ball screw actuator and gearbox, respectively, the desired actuator velocityqacan be calculated fromqby using (4.26).
1 3 3 3 3
3 3 2 3 3
(1/ ) 0
a 0
I I
q q (4.26)
MW‒AHRSv1 as IMU is used to measure the orientation of (ϕ, θ, ψ) and body angular velocity (ωx, ωy, ωz) of the cockpit, and the data is acquired by XBee module, at 100 Hz. The encoder with 10000 resolution is used to measure the position of the linear stage accurately. Then, the position (P0) and the velocity( )P0 of the cockpit are computed from the encoder through a low pass filter to minimize measurement noises. The filter is designed at 30Hz of the cutoff frequency and implemented to IMU and encoder. NI crio-9082 with 533 MHz real-time processor and 256 MB of DDR2RAM drives the control algorithm at 50Hz (0.02 sec) with six control modules NI-9512. Fig. 4.4 shows the SMP with the cockpit sphere as a flight simulator with a flight scene projected from the projector and flight instrument and control panels.
(a) (b)
Figure 4.4. Actual scaled SMP with full cockpit sphere. (a) SMP. (b) Inside of the cockpit.
Table 4.1. List of Specifications of Mechanical Parameters.
mC 150 kg Ia, It 77.4 kg•m2
rw / rs 0.0375 / 0.11 m RC 0.88 m
hs 0.35 m γ1 / γ2 5 mm / 60
qforward limit 0.285 m qreverse limit 0.745 m
, ( 1,2,3)
i sat
q i 0.5 m/s qi sat, ( 4,5,6)i 600 °/s
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The tracking control is numerically simulated in the presence of uncertainties with PID (4.27), I- SMC (4.29), and DOB-SMC (4.11), as shown in Figs. 4.6 and 4.7, and the results are compared to validate the performance against disturbances and uncertainties. Firstly, the cascade PID control in Fig.
4.5 is designed as (4.27).
Figure 4.5. PID control system for SMP.
2, 1, 2, 1, 2, 1,
1, 2, 2, 2, 2, 2, 2,
( ) ( ) ( )
( ) ( )
P P I P D P
P P I D I P D
k k k k dt k k
k k k dt k k k k
u e x e x e x
e e e x x x
(4.27)
where k1,P is a coefficient of outer P control; k2,P, k2,I, and k2,D are coefficients of inner P, I, and D control, respectively. The cascade PID controller removes the disturbances faster than conventional PID control but is still limited to handle numerous uncertainties.
For better performance, the I-SMC is applied to deal with the uncertainties. The sliding surface σ is defined in (4.28).
1 2
c c dt
σ e e e (4.28)
The corresponding control law is also designed as (4.29).
1 2
( , ) ( ) 1 sgn( )
A d c c 2 A D
u f x x M x e e M σ σ (4.29)
The stability of the I-SMC can be proved fromV12σ M σT A , (4.5), (4.28), and (4.29) asV= ‒ σT(ρDsgn(σ) ‒ ΔD) = |σ|(ρD‒bD) ≤ 0. The coefficients for each control method in Table 4.2 are
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determined from numerical simulations and then tuned by trial and error until good tracking performance is achieved.
Table 4.2. List of Specifications of Control Parameters.
DOB-SMC
λ0 3 λ1, λ2 1
k1* 2.8 (4) Li 10, i=1,2,‧‧‧,6
ρD 3 q* 0.833 (0.933)
c1 3.2 c2* 9 (4)
I-SMC
c1* 3.7 (3.2)
ρD 3
c2* 10 (6)
PID controller
k1,P* 1 (0.95) k2,P* 1 (0.8)
k2,I* 0.001 (0.001) k2,D* 0.006 (0.05)
*gain parameters for position (orientation) control
i) Comparison of Tracking Performance between simulations and experiments for DOB-SMC The numerical simulations are demonstrated for orientation tracking control of the cockpit using MATLAB. The uncertainties are extracted from the actual experimental data to make the simulations become reliable. The mechanical parameters for simulations are shown in Table 4.1. The control gains for each simulation are adjusted to maximize control performance. The desired trajectory is set as [30sin(0.1πt)° 0 0]T along roll motion from zero initial condition (ϕ0 = θ0 = ψ0 = 0°). For the control system, the kinematic uncertainties ΔK are estimated from the DOB in (4.8) and utilized for the DOB- SMC in (4.11) to reject ΔK as well as ΔD during actual operation. Fig. 4.6 shows the simulation results of tracking performance compared with the experiments. The DOB-SMC can estimate various uncertainties, and the tracking error in the finite-time is bounded within smaller than 2° in Fig. 4.6(a).
Moreover, DOB performance to estimate ΔK during the trajectory tracking is compared with both simulations and experiments. The estimation results are similar to each other, but the experiments in Fig. 4.6(b) are larger than the simulation since the SMP has a number of uncertainties such as distorted cockpit sphere, surface contact, and slipping, unlike the simulations. The results can show the feasibility of estimation and tracking performance of the DOB-SMC regardless of the presence of the uncertainties validating the kinematic and dynamic models.
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(b)
Figure 4.6. Performance comparison of simulation and experimental results for DOB-SMC. (a) Tracking results. (b) Results of estimation of kinematic uncertainties.
ii) Simulation for Robustness Validation of DOB-SMC
The robustness of the DOB-SMC is shown with numerical simulations. The cockpit is initially set as [20° 0 0]T with an angular velocity of [10°/s 0 0]T. The uncertainties ΔK = [10 10 10]T °/s and ΔD
= [6 6 6]T Nm are injected to each simulation after 10 sec. Fig. 4.7 shows the tracking performance of the DOB-SMC compared with the PID and I-SMC. The DOB-SMC tracks more accurately than both I-SMC and PID in the presence of kinematic and dynamic uncertainties.
Similarly, the effects of uncertain parameters such as frictional torque and inertia on dynamics are investigated. The frictional torque is varying to 3sgn( )Nm due to the changing of inertia 10 kg•m2 for human on board. The step input of 30° is applied for roll motion with and without uncertainties. Fig.
4.8 shows the performance of the controllers. The frictional torque dramatically affects the control
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-40 0 40
0 10 20 30 40 50 60
-4 0 4
0 10 20 30 40 50 60
-2 0 2
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performance of both PID and I-SMC, but the DOB-SMC shows robustness against the parameter uncertainties.
(a) (b)
Figure 4.7. Effects of constant uncertainties. (a) Kinematic uncertainties. (b) Dynamic uncertainties.
(a) (b)
Figure 4.8. Effects of parameter uncertainties. (a) Uncertain friction. (b) Uncertain inertia. (F) and (I) represent the presence of uncertain friction torque and inertia.
0 5 10 15 20
0 10 20 30 40
. . .
0 5 10 15 20
0 10 20 30 40
. . .
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4.4.1.2 Experiments for Orientation and Position Control
The experimental results for PID, I-SMC, and DOB-SMC are compared using two different inputs:
step and sinusoidal inputs. The comparison demonstrates the tracking performance of the proposed control system. The gain parameters for each controller in orientation and position tracking control, as shown in Table 4.2, are determined by trial and error until good tracking performances are acquired from the basis of simulation results.
i) Orientation Control
Two inputs for tracking control of the cockpit are step and sinusoidal inputs; [ϕd θd ψd]T = [60 0 0]T and [30sin(0.1πt) 0 0]T, respectively. Initial orientation set as zero of ϕ0, θ0, and ψ0. The actual control input for each controller can be generated from (4.26). The tracking performance of three controllers are compared in Fig. 4.9 and 4.10. Firstly, for the step response (see Fig. 4.9), the maximum error (eMax) of the DOB-SMC is 0.09° along all axes, but 0.73°, PID and 0.12°, I-SMC after 30sec. The root-mean-square error (RMSE) of the DOB-SMC, I-SMC, and PID after 30sec is 0.072°, 0.074°, and 0.22°, respectively. The results are detailed in Table 4.3. Moreover, the error converges in finite-time less than 10 sec in the DOB-SMC compared to 30sec and 15sec in PID and I-SMC, respectively. The results show that the DOB-SMC has faster convergence and smaller steady-state error than PID and I- SMC in the presence of uncertainties.
Next, the sinusoidal input is applied for 60 sec, as shown in Fig. 4.10. eMax of the DOB-SMC is 1.54°
along all axes while 10.88° and 3.35° from PID and I-SMC, respectively, for 60 sec. The RMSE of the DOB-SMC is 0.65° smaller than 5.43°, PID and 1.94°, I-SMC. The results indicate better orientation tracking performance. Moreover, computational time causes time delays between the input command and the output in the control loop. The DOB-SMC takes less than 0.1 sec since it guarantees finite-time convergence in spite of uncertainties with smaller delays compared to PID (≈ 1sec) and I-SMC (≈
0.2sec).
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(a) (b)
(c) (d)
(e) (f)
Figure 4.9. Comparison of orientation tracking control results for step input (ϕd = 60°, θd, ψd = 0°).
(a), (b) PID. (c), (d) I-SMC. (e), (f) DOB-SMC.
Table 4.3. Comparison of Orientation Control Performance for PID, I-SMC, and DOB-SMC.
Controller ϕ θ ψ
Step Input
PID 0.22/0.53 0.19/0.56 0.21/0.73
I-SMC 0.052/0.12 0.038/0.06 0.074/0.1
DOB-SMC 0.026/0.051 0.015/0.04 0.072/0.09
Sinusoidal Input
PID 5.43/10.88 2.65/6.50 2.83/7.31
I-SMC 1.46/2.86 1.94/3.35 0.40/0.86
DOB-SMC 0.65/1.54 0.53/1.47 0.22/0.61
RMSE/eMax (°)
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0 20 40 60
0 10 20 30 40 50 60
0 20 40 60
0 10 20 30 40 50 60
0 20 40 60
0 10 20 30 40 50 60
0 20 40 60
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(a) (b)
(c) (d)
(e) (f)
Figure 4.10. Comparison of orientation tracking control results for sinusoidal input (ϕd = 30sin(0.1πt)°, θd, ψd = 0°). (a), (b) PID. (c), (d) I-SMC. (e), (f) DOB-SMC.
Control efforts defined as iN1
|| ( )||u i 2t i( )
are computed where N is the number of total sampling steps in the experiments. Δt(i) is a time interval at the ith step. Fig. 4.11 shows the results from three controllers for the step response in Fig. 4.9. The I-SMC and DOB-SMC need much smaller control efforts than PID. Besides, the DOB-SMC consumes the effort three times and forty times smaller than I-SMC in both transient and steady states, respectively, since the uncertainties are effectively compensated and rejected from DOB estimation.0 10 20 30 40 50 60
-40 -20 0 20 40
0 10 20 30 40 50 60
-10 -5 0 5 10
0 10 20 30 40 50 60
-40 -20 0 20 40
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Figure 4.11. Control efforts for PID, I-SMC, and DOB-SMC.
ii) Position Control
The translational motion generated by the ball screw actuators has a mechanical limitation that is forward and reverse limit for safety. Then, the position of each actuator qi must satisfy the following inequality, qforward limit ≤ qi ≤ qreverse limit, i=1,2,3, where the limit position is detailed in Table 4.1. The actuator input for each controller can also be generated from (4.26). When the cockpit moves along the translational axis within the motion range, rotation can occur due to kinematic coupling. Thus, six DOFs motion should be controlled. A desired command input is given as [(P0,X)d (P0,Y)d (P0,Z)d]T = [0.15sin(0.04πt) 0.15cos(0.04πt) 0]T m and [ϕd θd ψd]T = [0 0 0]T with zero initial conditions for both position and orientation, respectively. Fig. 4.12 shows the comparison of position and orientation tracking control along the circular path with PID, I-SMC, and DOB-SMC. The eMax in the DOB-SMC is 0.008m along all axes, smaller than 0.032m and 0.01m in PID and I-SMC, respectively. The RMSE of the DOB-SMC is 0.0056 m smaller than 0.023 m, PID and 0.0069m, I- SMC. Similarly, it is validated that the undesired coupled rotation during the translation is canceled out from the DOB-SMC. The eMax is about 1° in the DOB-SMC smaller than 4° in PID and 3° in I-SMC in Figs. 4.12(c), (f), and (i). The RMSE in the DOB-SMC is 0.7° smaller than 2.9° and 1.5° in PID and I- SMC, respectively. The detailed results are summarized in Table 4.4. The experimental results show that the orientation and position tracking performance of the DOB-SMC is better than the PID and I- SMC even in the presence of uncertainties similar to the numerical simulation.
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 4.12. Comparison of position and orientation tracking control results. (a), (b), (c) PID. (d), (e), (f) I-SMC. (g), (h), (i) DOB-SMC.
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Table 4.4. Comparison of Position and Orientation Control Performance for PID, I-SMC, and DOB-SMC.
position
controller X Y Z
PID 2.1e−2/3.2e−2 2.3e−2/3.2e−2 5e−4/7e−4
I-SMC 6.9e−3/1.0e−2 1.6e−2/1.0e−2 3.2e−4/5e−4
DOB-SMC 5.2e−3/8e−3 5.6e−3/8e−3 3e−4/5e−4
coupled rotation (°)
controller ϕ θ ψ
PID 2.91/4.41 2.25/4.02 0.55/1.59
I-SMC 1.53/3.67 1.22/2.04 0.11/0.31
DOB-SMC 0.64/1.37 0.58/0.89 0.21/0.42
RMSE/eMax
iii) Experiments of Flight Maneuvers
Actual flight simulations are demonstrated using the SMP for VR. Two typical flight maneuvers are explored as phugoid and dutch roll modes related to lateral and longitudinal motion, respectively. The flight dynamics are utilized to obtain the trajectory based on the real flight to increase the reality detailed in [62]. The dynamic equations are solved from MATLAB to generate the trajectories for the SMP. The rotational motion is fully driven without any scaling and limiting from the advantages of the SMP with unlimited rotation. In contrast to translational motion, high pass filter, as well as scaling and limiting, are used for the position trajectory due to the limitation of motion range.
iii-1) Phugoid mode
The phugoid mode is an oscillating motion along a longitudinal direction and pitch axis with a lightly damped low frequency, as shown in Fig. 4.13(a). Figs. 4.13(b)‒(e) show the tracking results for experiments of the phugoid motion, including position and orientation. The tracking performance is demonstrated from the results for the trajectory using DOB-SMC within 1.3° of eMax.
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(b) (c)
(d) (e)
Figure 4.13. Experimental results for phugoid mode. (a) Side view. (b), (c) – Tracking performance for position. (d), (e) – Tracking performance for orientation.
iii-2) Dutch Roll Mode
The dutch roll mode is one of the classical dynamic responses of flight, a complex interaction between three lateral-directional DOFs motion composed of sideslip, rolling, and yawing. The oscillatory conversion between rolling and yawing occurs, as shown in Figs. 4.14(a) and (b). Figs.
4.14(c)‒(f) show tracking results for experiments of the dutch roll mode coupled between three DOFs motion. The tracking performance is validated from the experimental results for the desired trajectory, applying proposed control within the eMax of 6° initially. The error converges after 50 sec within 1° in spite of highly coupled motion. The results show accurate tracking performance to simulate actual flight motion successfully, and the SMP and DOB-SMC can be applied to the flight simulator for VR applications.
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-0.1 -0.05 0 0.05 0.1
0 25 50 75 100
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(a) (b)
(c) (d)
(e) (f)
Figure 4.14. Representation and experimental results for dutch roll mode. (a) Front view. (b) Top view. (c), (d) – Tracking performance for position. (e), (f) – Tracking performance for orientation.
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